Here is the full list of all 21 Rooms: The Unsolvable Puzzle achievements.
Make the toymaker's dream come true.
Don't worry about failing.
Escape the toymaker's mansion.
Earn 432 puzzle pieces.
Clear all underground levels in Mansion 1.
Destroy as many puppets as you can.
Clear all underground levels in Mansion 4.
Clear all underground levels in Mansion 3.
Clear all underground levels in Mansion 2.
Earn 324 puzzle pieces.
Clear all levels in Mansion 4.
Rotate rooms often.
Use the lamp often.
Earn 216 puzzle pieces.
Clear all levels in Mansion3.
Clear all levels in Mansion 2.
Earn 130 puzzle pieces.
Move rooms often.
Earn 80 puzzle pieces.
Clear all levels in Mansion 1.
Earn 30 puzzle pieces.